In this note we study common preperiodic points of rational maps of the Riemann Sphere. We show that given any degrees \(d_1,d_2\ge 2\) , outside a Zariski closed subset of the space of pairs of rational maps (f, g) of degree \(d_1\) and \(d_2\) respectively, the maps f and g share at most a uniformly bounded number of common preperiodic points. This generalizes a result of DeMarco and Mavraki to maps of possibly different degrees. Our main contribution is the use of Hölder properties of the Green function of a rational map to obtain height estimates.